1) An infinite sequence of integers has an infinite number of elements.
2) For $\lim \limits_{n \to \infty} f(n)$, I prefer to say the limit as $n$ grows without bound.
3) It's false whatever the context is. 0.4999... = 0.5000... end of story.
4) I suggest you read your own post (#1) to see what you wrote.
5) No it isn't. It clearly has certain properties that your $P_n$ idea denies, such as having an infinite number of elements. If you wish to deny the existence of infinite sequences, you can no longer talk about Cantor's argument because there existence is part of the system in which he was working.

Before you start claiming anything about your $P_n$ idea, you need to look at post #9 in particular.

You keep mentioning this, apparently with a view to justifying (eventually) your disagreement with it. However, you have never been able to state at precisely what stage Cantor makes any mistake, instead making somewhat vague claims along the lines of "he can't do that because...", but then writing about something that Cantor didn't do anyway.

There is no point in defining something as a limit if you don't explain what you mean by "limit" in your particular context. Doing so makes your mathematical arguments imprecise and hence your conclusions are unproved.

I addressed Cantor's argument a million times, in addition to all the other proofs of uncountability, in previous threads. In brief, any rational, intelligent discussion about properties at infinity has to begin with a discussion of finite properties at n, and if the conclusion holds for all n it holds for n -> infinity. That is absolutely fundamental.

How do you propose to do this? I don't see any way this can be achieved. The number of zeros is finite and must remain finite in order to have anything following them. It never gets any closer to being infinite and thus fails to "approach infinity".

Quote:

Originally Posted by zylo

Limit 1: Representation Limit.
Each member of the list remains unique and becomes a real number.

How does this happen? Each number already is a real number. I have already discussed the impossibility of the representations becoming infinite, or even of being infinite if they end with particular digits.

Quote:

Originally Posted by zylo

Limit 2: Analytic Limit .
Each member of the list equals .3

This isn't true either, although every infinite sequences composed only of members of your lists converges to 0.3. That just shows that 0.3 is a real number. I don't think anyone is reeling in shock at that.

Quote:

Originally Posted by zylo

Without distinguishing the limits, you have the impossible situation that the decimal representation of any number is not unique as n approaches infinity.

The fallacy here is again your meaningless phrase "as $n$ approaches infinity". It doesn't and it can't. Thetwo real numbers represented by the infinite decimals $0.a_1 a_2 a_3 \ldots$ and $0.b_1 b_2 b_3 \ldots$ are distinct if the limits $a = \lim \limits_{n \to \infty} \sum \limits_{k=1}^n \frac{a_k}{10^n}$ and $b = \lim \limits_{n \to \infty} \sum \limits_{k=1}^n \frac{b_k}{10^n}$ are distinct. (Note that the notation $n \to \infinity$ should be read as "as $n$ grows without bound" and that each $a_k$ and each $b_k$ is a decimal numeral between zero and nine inclusive).

There is a very difficult philosophical and conceptual problem here which is that we define a meaning for $\sum_{n=1}^\infty \frac{a_1}{10^n}$ without (apparently) knowing for sure how to compute an infinite sum. However:

If the infinite sum does have a meaningful value, subject to certain axioms such as, perhaps, that the addition of infinite sums is a linear operation, the value is the one defined by the limit;

the definition of the reals as the closure of infinite Cauchy sequences of rationals is equivalent to the definition of reals as being decimal numbers.

Quote:

Originally Posted by zylo

any rational, intelligent discussion about properties at infinity has to begin with a discussion of finite properties at n, and if the conclusion holds for all n it holds for n -> infinity. That is absolutely fundamental.

If by "n -> infinity", you mean "as n grows without bound", this is trivially true.
But since you are trying to say that what holds for every finite case must hold for the infinite case, it is trivially false because the finite case is finite and the infinite case is infinite. So we immediately have a contradiction.

Limit 1: Representation Limit.
Each member of the list remains unique and becomes a real number.

If you're concentrating on them as just representations, they're not numbers, just representations, and don't become real numbers. On the other hand, if they are numbers to start with, they're already real numbers and therefore they can't "become real numbers".

Quote:

Originally Posted by zylo

Limit 2: Analytic Limit.
Each member of the list equals .3

Each member of what list? You are not defining either usage of "limit".

Quote:

Originally Posted by zylo

Without distinguishing the limits, you have the impossible situation that the decimal representation of any number is not unique as n approaches infinity.

You're not making yourself clear. What does "as n approaches infinity" mean in this context, and how can something be impossible if you've given an example of it? In most articles, the distinction you're making isn't relevant, as the context in which uniqueness applies is properly explained. Also, the decimal representation of rationals isn't unique. For example, the same value, 1, can be represented as 1, 1.0, 1.00, 0.999... (0.9 recurring), etc.

Quote:

Originally Posted by zylo

Mathematicians fail to make the distinction which leads to unnecessary complication and confusion.

That wouldn't apply to Cantor's proofs that you've discussed, as Cantor is clearly writing about sequences, and doesn't use words such as "real", "limit" or "represents". On the other hand, you do use such words, and often fail to draw attention to the distinction you're making now. Also, you haven't defined what you meant by "limit" in various contexts where you used that word, even when I've specifically asked you to do so.

So you agree that .499... and .5 both equal 1/2, which means decimal representation of 1/2 isn't unique?

Quote:

Originally Posted by zylo

In brief, any rational, intelligent discussion about properties at infinity has to begin with a discussion of finite properties at n, and if the conclusion holds for all n it holds for n -> infinity.

If "for n -> infinity" refers to a limit, you have failed to define that limit properly. Also, when a sequence has a properly-defined limit, it is typically the case that the limit has properties that aren't possessed by individual members of the sequence. For example, pi is the limit of various sequences of rationals, but pi isn't a rational.